Moment Generating Function of the Age of Gossip in Networks

We study a general setting of gossip networks in which a source node forwards its measurements (in the form of status updates) about some observed physical process to a set of monitoring nodes according to independent Poisson processes. Further, each monitoring node sends status updates about its information status (about the process observed by the source) to the other monitoring nodes according to independent Poisson processes. We quantify the freshness of information available at each monitoring node in terms of Age of Information (AoI). While this setting has been analyzed in a handful of prior works, the focus has been on characterizing the average (i.e., marginal first moment) of each age process. In contrast, our analysis is focused on understanding the distributional properties of the AoI processes through the characterization of their stationary marginal and joint moment generating functions (MGFs). In particular, for the serially and parallelly-connected gossip network topologies, we derive closed-form expressions for marginal/joint higher-order statistics of age processes, such as the variance of each age process and the correlation coefficients between all possible pairwise combinations of age processes. Our analytical results demonstrate the importance of incorporating the higher-order moments of age processes in the implementation/optimization of age-aware gossip networks rather than just relying on their average values.